A309287 Square array T(v, m), read by antidiagonals, for the Rogel-Klee arithmetic function: number of positive integers h in the set [m] for which gcd(h, m) is v-th-power-free, i.e., gcd(h, m) is not divisible by any v-th power of an integer > 1 (with v, m >= 1).

1, 1, 1, 2, 2, 1, 2, 3, 2, 1, 4, 3, 3, 2, 1, 2, 5, 4, 3, 2, 1, 6, 6, 5, 4, 3, 2, 1, 4, 7, 6, 5, 4, 3, 2, 1, 6, 6, 7, 6, 5, 4, 3, 2, 1, 4, 8, 7, 7, 6, 5, 4, 3, 2, 1, 10, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 4, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 12, 9, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 6, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1

Offset: 1

Author

Petros Hadjicostas, Jul 21 2019

Keywords

• nonn
• tabl

Comments

For fixed v >= 1, T(v, .) is a multiplicative arithmetic function with T(v, 1) = 1; T(v, p^e) = p^e, if e < v; and T(v, p^e) = p^e - p^(e-v) if e >= v (where p is a prime >= 2).

Here, T(v=1, m) = phi(m) is the number of arithmetic progressions (s*m + k: s >= 0), k = 1, ..., m, that contain infinitely many primes (by Dirichlet's theorem). For v >= 2, T(v, m) is the number of these arithmetic progressions that contain infinitely many v-th-power-free numbers.

In Section 6 of his paper, Cohen (1959) mentions that this function was introduced by Rogel (1900) in an article published in a Bohemian journal. Roger's (1900) paper is a continuation of Rogel (1897) and the two should be read together.

McCarthy (1958) uses the asymptotic result given in the FORMULA section below to prove that the probability that the GCD of two positive integers is v-th-power-free is 1/zeta(2*v).

Examples

			Table for T(v, m) (with rows v >= 1 and columns m >= 1) begins as follows:
  v=1: 1, 1, 2, 2, 4, 2, 6, 4, 6,  4, 10,  4, 12,  6,  8,  8, ...
  v=2: 1, 2, 3, 3, 5, 6, 7, 6, 8, 10, 11,  9, 13, 14, 15, 12, ...
  v=3: 1, 2, 3, 4, 5, 6, 7, 7, 9, 10, 11, 12, 13, 14, 15, 14, ...
  v=4: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 15, ...
  v=5: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, ...
  v=6: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, ...
  v=7: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, ...
  ...
Clearly, lim_{v -> infinity} T(v, m) = m.
		

References

• Paul J. McCarthy, Introduction to Arithmetical Functions, Springer-Verlag, 1986; see pp. 38-40 and 69.

Links

• Eckford Cohen, A class of residue systems (mod r) and related arithmetical functions. I. A generalization of the Moebius function, Pacific J. Math. 9(1) (1959), 13-24; see Section 6 where T(v, m) = Phi_v(m).
• Eckford Cohen, A generalized Euler phi-function, Math. Mag. 41 (1968), 276-279; here T(v, m) = phi_v(m).
• E. K. Haviland, An analogue of Euler's phi-function, Duke Math. J. 11 (1944), 869-872; here T(v=2, m) = rho(m).
• V. L. Klee, Jr., A generalization of Euler's phi function, Amer. Math. Monthly, 55(6) (1948), 358-359; here T(v, m) = Phi_v(m).
• Paul J. McCarthy, On a certain family of arithmetic functions, Amer. Math. Monthly 65 (1958), 586-590; here, T(v, m) = T_v(m).
• Franz Rogel, Entwicklung einiger zahlentheoreticher Funktionen in unendliche Reihen, S.-B. Kgl. Bohmischen Ges. Wiss. Article XLVI/XLIV (1897), Prague (26 pages). [This paper deals with arithmetic functions, especially the Euler phi function. It was continued three years later with the next paper, which contains his function phi_k(n). As stated at the end of the volume, in the table of contents, there is a mistake in numbering the article, so two Roman numerals appear in the literature for labeling this article!]
• Franz Rogel, Entwicklung einiger zahlentheoreticher Funktionen in unendliche Reihen, S.-B. Kgl. Bohmischen Ges. Wiss. Article XXX (1900), Prague (9 pages). [This is a continuation of the previous article, which was written three years earlier and has the same title. The numbering of the equations continues from the previous paper, but this paper is the one that introduces the function phi_k(n). In our notation, T(v, m) = phi_v(m). Cohen (1959) refers to this paper and correctly attributes this function to F. Rogel.]

Crossrefs

A000010 (row v = 1 is Euler's phi function), A063659 (row v = 2 is Haviland's function), A254926 (row v = 3).

Programs

PARI

/* Modification of Michel Marcus's program from sequence A254926: */
T(v, m) = {f = factor(m); for (i=1, #f~, if ((e=f[i, 2])>=v, f[i, 1] = f[i, 1]^e - f[i, 1]^(e-v); f[i, 2]=1); ); factorback(f); }
/* Print the first 40 terms of each of the first 10 rows: */
{ for (v=1, 10, for (m=1, 40, print1(T(v, m), ", "); ); print(); ); }

Formula

T(v, m) = m * Product_{p prime and p^v|m} (1 - p^(-v)) for v, m >= 1.

T(v, m) = Sum_{n >= 1} mu(n) * [m, n^v] * (m/n^v), where [m, n^v] = 1 when m is a multiple of n^v, and = 0 otherwise. [This is Eq. (53) in Rogel (1900) and Eq. (6.1) in Cohen (1959).]

Dirichlet g.f. for row v: Sum_{m >= 1} T(v, m)/m^s = zeta(s-1)/zeta(v*s) for Re(s) > 1.

Asymptotics: Sum_{m = 1..n} T(v, m) = n^2/(2*zeta(2*v)) + O(n) for v >= 2 and = n^2/(2*zeta(2)) + O(n*log(n)) for v = 1 (for Euler's phi-function).

Analog of Fermat's theorem: if gcd(a, m) = 1 with a >= 1, then m/gcd(a^T(v, m) - 1, m) is v-th-power-free. (For v = 1, this means m/gcd(a^T(v=1, m) - 1, m) = 1.)

T(v, m^v)/m^v = Sum_{d|m} mu(d)/d^v for m, v >= 1. (It generalizes the formula phi(m)/m = Sum_{d|m} mu(d)/d since phi(m) = T(v=1, m).)